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A prompt is a sequence of symbol or tokens, selected from a vocabulary according to some rule, which is prepended/concatenated to a textual query. A key problem is how to select the sequence of tokens: in this paper we formulate it as a combinatorial optimization problem. The high dimensionality of the token space com-pounded by the length of the prompt sequence requires a very efficient solution. In this paper we propose a Bayesian optimization method, executed in a continuous em-bedding of the combinatorial space. In this paper we focus on hard prompt tuning (HPT) which directly searches for discrete tokens to be added to the text input with-out requiring access to the large language model (LLM) and can be used also when LLM is available only as a black-box. This is critically important if LLMs are made available in the Model as a Service (MaaS) manner as in GPT-4. The current manu-script is focused on the optimization of discrete prompts for classification tasks. The discrete prompts give rise to difficult combinatorial optimization problem which easily become intractable given the dimension of the token space in realistic applications. The optimization method considered in this paper is Bayesian optimization (BO) which has become the dominant approach in black-box optimization for its sample efficiency along with its modular structure and versatility. In this paper we use BoTorch, a library for Bayesian optimization research built on top of pyTorch. Albeit preliminary and obtained using a 'vanilla' version of BO, the experiments on RoB-ERTa on six benchmarks, show a good performance across a variety of tasks and enable an analysis of the tradeoff between size of the search space, accuracy and wall clock time.

Gaussian Process regression is a kernel method successfully adopted in many real-life applications. Recently, there is a growing interest on extending this method to non-Euclidean input spaces, like the one considered in this paper, consisting of probability measures. Although a Positive Definite kernel can be defined by using a suitable distance -- the Wasserstein distance -- the common procedure for learning the Gaussian Process model can fail due to numerical issues, arising earlier and more frequently than in the case of an Euclidean input space and, as demonstrated in this paper, that cannot be avoided by adding artificial noise (nugget effect) as usually done. This paper uncovers the main reason of these issues, that is a non-stationarity relationship between the Wasserstein-based squared exponential kernel and its Euclidean-based counterpart. As a relevant result, the Gaussian Process model is learned by assuming the input space as Euclidean and then an algebraic transformation, based on the uncovered relation, is used to transform it into a non-stationary and Wasserstein-based Gaussian Process model over probability measures. This algebraic transformation is simpler than log-exp maps used in the case of data belonging to Riemannian manifolds and recently extended to consider the pseudo-Riemannian structure of an input space equipped with the Wasserstein distance.

Optimal resource allocation is gaining a renewed interest due its relevance as a core problem in managing, over time, cloud and high-performance computing facilities. Semi-Bandit Feedback (SBF) is the reference method for efficiently solving this problem. In this paper we propose (i) an extension of the optimal resource allocation to a more general class of problems, specifically with resources availability changing over time, and (ii) Bayesian Optimization as a more efficient alternative to SBF. Three algorithms for Bayesian Optimization for Resource Allocation, namely BORA, are presented, working on allocation decisions represented as numerical vectors or distributions. The second option required to consider the Wasserstein distance as a more suitable metric to use into one of the BORA algorithms. Results on (i) the original SBF case study proposed in the literature, and (ii) a real-life application (i.e., the optimization of multi-channel marketing) empirically prove that BORA is a more efficient and effective learning-and-optimization framework than SBF., Comment: 31 pages, 12 figures

There is a consensus that focusing only on accuracy in searching for optimal machine learning models amplifies biases contained in the data, leading to unfair predictions and decision supports. Recently, multi-objective hyperparameter optimization has been proposed to search for machine learning models which offer equally Pareto-efficient trade-offs between accuracy and fairness. Although these approaches proved to be more versatile than fairness-aware machine learning algorithms -- which optimize accuracy constrained to some threshold on fairness -- they could drastically increase the energy consumption in the case of large datasets. In this paper we propose FanG-HPO, a Fair and Green Hyperparameter Optimization (HPO) approach based on both multi-objective and multiple information source Bayesian optimization. FanG-HPO uses subsets of the large dataset (aka information sources) to obtain cheap approximations of both accuracy and fairness, and multi-objective Bayesian Optimization to efficiently identify Pareto-efficient machine learning models. Experiments consider two benchmark (fairness) datasets and two machine learning algorithms (XGBoost and Multi-Layer Perceptron), and provide an assessment of FanG-HPO against both fairness-aware machine learning algorithms and hyperparameter optimization via a multi-objective single-source optimization algorithm in BoTorch, a state-of-the-art platform for Bayesian Optimization.

The main objective of this paper is to outline a theoretical framework to characterise humans' decision-making strategies under uncertainty, in particular active learning in a black-box optimization task and trading-off between information gathering (exploration) and reward seeking (exploitation). Humans' decisions making according to these two objectives can be modelled in terms of Pareto rationality. If a decision set contains a Pareto efficient strategy, a rational decision maker should always select the dominant strategy over its dominated alternatives. A distance from the Pareto frontier determines whether a choice is Pareto rational. To collect data about humans' strategies we have used a gaming application that shows the game field, with previous decisions and observations, as well as the score obtained. The key element in this paper is the representation of behavioural patterns of human learners as a discrete probability distribution. This maps the problem of the characterization of humans' behaviour into a space whose elements are probability distributions structured by a distance between histograms, namely the Wasserstein distance (WST). The distributional analysis gives new insights about human search strategies and their deviations from Pareto rationality. Since the uncertainty is one of the two objectives defining the Pareto frontier, the analysis has been performed for three different uncertainty quantification measures to identify which better explains the Pareto compliant behavioural patterns. Beside the analysis of individual patterns WST has also enabled a global analysis computing the barycenters and WST k-means clustering. A further analysis has been performed by a decision tree to relate non-Paretian behaviour, characterized by exasperated exploitation, to the dynamics of the evolution of the reward seeking process., Comment: 49 pages, 39 figures. arXiv admin note: substantial text overlap with arXiv:2102.07647

Optimal sensor placement (SP) usually minimizes an impact measure, such as the amount of contaminated water or the number of inhabitants affected before detection. The common choice is to minimize the minimum detection time (MDT) averaged over a set of contamination events, with contaminant injected at a different location. Given a SP, propagation is simulated through a hydraulic software model of the network to obtain spatio-temporal concentrations and the average MDT. Searching for an optimal SP is NP-hard: even for mid-size networks, efficient search methods are required, among which evolutionary approaches are often used. A bi-objective formalization is proposed: minimizing the average MDT and its standard deviation, that is the risk to detect some contamination event too late than the average MDT. We propose a data structure (sort of spatio-temporal heatmap) collecting simulation outcomes for every SP and particularly suitable for evolutionary optimization. Indeed, the proposed data structure enabled a convergence analysis of a population-based algorithm, leading to the identification of indicators for detecting problem-specific converge issues which could be generalized to other similar problems. We used Pymoo, a recent Python framework flexible enough to incorporate our problem specific termination criterion. Results on a benchmark and a real-world network are presented., Comment: 9 pages, 15 figures

This paper addresses black-box optimization over multiple information sources whose both fidelity and query cost change over the search space, that is they are location dependent. The approach uses: (i) an Augmented Gaussian Process, recently proposed in multi-information source optimization as a single model of the objective function over search space and sources, and (ii) a Gaussian Process to model the location-dependent cost of each source. The former is used into a Confidence Bound based acquisition function to select the next source and location to query, while the latter is used to penalize the value of the acquisition depending on the expected query cost for any source-location pair. The proposed approach is evaluated on a set of Hyperparameters Optimization tasks, consisting of two Machine Learning classifiers and three datasets of different sizes.

The main objective of this paper is to outline a theoretical framework to analyse how humans' decision-making strategies under uncertainty manage the trade-off between information gathering (exploration) and reward seeking (exploitation). A key observation, motivating this line of research, is the awareness that human learners are amazingly fast and effective at adapting to unfamiliar environments and incorporating upcoming knowledge: this is an intriguing behaviour for cognitive sciences as well as an important challenge for Machine Learning. The target problem considered is active learning in a black-box optimization task and more specifically how the exploration/exploitation dilemma can be modelled within Gaussian Process based Bayesian Optimization framework, which is in turn based on uncertainty quantification. The main contribution is to analyse humans' decisions with respect to Pareto rationality where the two objectives are improvement expected and uncertainty quantification. According to this Pareto rationality model, if a decision set contains a Pareto efficient (dominant) strategy, a rational decision maker should always select the dominant strategy over its dominated alternatives. The distance from the Pareto frontier determines whether a choice is (Pareto) rational (i.e., lays on the frontier) or is associated to "exasperate" exploration. However, since the uncertainty is one of the two objectives defining the Pareto frontier, we have investigated three different uncertainty quantification measures and selected the one resulting more compliant with the Pareto rationality model proposed. The key result is an analytical framework to characterize how deviations from "rationality" depend on uncertainty quantifications and the evolution of the reward seeking process.

Searching for accurate Machine and Deep Learning models is a computationally expensive and awfully energivorous process. A strategy which has been gaining recently importance to drastically reduce computational time and energy consumed is to exploit the availability of different information sources, with different computational costs and different "fidelity", typically smaller portions of a large dataset. The multi-source optimization strategy fits into the scheme of Gaussian Process based Bayesian Optimization. An Augmented Gaussian Process method exploiting multiple information sources (namely, AGP-MISO) is proposed. The Augmented Gaussian Process is trained using only "reliable" information among available sources. A novel acquisition function is defined according to the Augmented Gaussian Process. Computational results are reported related to the optimization of the hyperparameters of a Support Vector Machine (SVM) classifier using two sources: a large dataset - the most expensive one - and a smaller portion of it. A comparison with a traditional Bayesian Optimization approach to optimize the hyperparameters of the SVM classifier on the large dataset only is reported., Comment: 22 pages, 4 figures, submitted to Soft computing - Special Issue on "Optimization methods for decision making: advances and applications"

Resilience is meant as the capability of a networked infrastructure to provide its service even if some components fail: in this paper we focus on how resilience depends both on net-wide measures of connectivity and the role of a single component. This paper has two objectives: first to show how a set of global measures can be obtained using techniques from network theory, in particular how the spectral analysis of the adjacency and Laplacian matrices and a similarity measure based on Jensen-Shannon divergence allows us to obtain a characteriza-tion of global connectivity which is both mathematically sound and operational. Second, how a clustering method in the subspace spanned by the l smallest eigen-vectors of the Laplacian matrix allows us to identify the edges of the network whose failure breaks down the network. Even if most of the analysis can be applied to a generic networked infrastructure, specific references will be made to Water Distribution Networks (WDN)., Comment: 14 pages, 4 figures, submitted to Network - Special Issue on "Network based system optimization"

Modelling human function learning has been the subject of in-tense research in cognitive sciences. The topic is relevant in black-box optimization where information about the objective and/or constraints is not available and must be learned through function evaluations. In this paper we focus on the relation between the behaviour of humans searching for the maximum and the probabilistic model used in Bayesian Optimization. As surrogate models of the unknown function both Gaussian Processes and Random Forest have been considered: the Bayesian learning paradigm is central in the development of active learning approaches balancing exploration/exploitation in uncertain conditions towards effective generalization in large decision spaces. In this paper we analyse experimentally how Bayesian Optimization compares to humans searching for the maximum of an unknown 2D function. A set of controlled experiments with 60 subjects, using both surrogate models, confirm that Bayesian Optimization provides a general model to represent individual patterns of active learning in humans

Sequential Model-based Bayesian Optimization has been successful-ly applied to several application domains, characterized by complex search spaces, such as Automated Machine Learning and Neural Architecture Search. This paper focuses on optimal control problems, proposing a Sequential Model-based Bayesian Optimization framework to learn optimal control strategies. A quite general formalization of the problem is provided, along with a specific instance related to optimization of pumping operations in an urban Water Distri-bution Network. Relevant results on a real-life Water Distribution Network are reported, comparing different possible choices for the proposed framework.

Bayesian Optimization has become the reference method for the global optimization of black box, expensive and possibly noisy functions. Bayesian Op-timization learns a probabilistic model about the objective function, usually a Gaussian Process, and builds, depending on its mean and variance, an acquisition function whose optimizer yields the new evaluation point, leading to update the probabilistic surrogate model. Despite its sample efficiency, Bayesian Optimiza-tion does not scale well with the dimensions of the problem. The optimization of the acquisition function has received less attention because its computational cost is usually considered negligible compared to that of the evaluation of the objec-tive function. Its efficient optimization is often inhibited, particularly in high di-mensional problems, by multiple extrema. In this paper we leverage the addition-ality of the objective function into mapping both the kernel and the acquisition function of the Bayesian Optimization in lower dimensional subspaces. This ap-proach makes more efficient the learning/updating of the probabilistic surrogate model and allows an efficient optimization of the acquisition function. Experi-mental results are presented for real-life application, that is the control of pumps in urban water distribution systems.